The big idea: A function is increasing on an interval when the graph goes up as you move from left to right.
It is decreasing when the graph goes down.
The key word: The key word is interval — we describe the range of x-values, not individual points.
- Increasing interval
- A range of x-values where f(x) gets larger as x increases. The graph rises left to right.
- Decreasing interval
- A range of x-values where f(x) gets smaller as x increases. The graph falls left to right.
- Stationary point
- A point where the graph is neither increasing nor decreasing — the gradient is zero.
Increasing (↗)
- Graph rises as you move left to right
- f(x) gets larger as x increases
- Happens before a local maximum and after a local minimum
- Express as: x < a or x > b
Decreasing (↘)
- Graph falls as you move left to right
- f(x) gets smaller as x increases
- Happens between a local maximum and the next minimum
- Express as: a < x < b
Intervals, not points: IB asks for intervals, not individual x-values.
Your answer must be a range such as 'x > 2' or '1 < x < 4'.
Saying 'x = 2' describes a single point — that is not an interval.
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The big idea: Find where the graph changes direction — those are the turning points.
The increasing and decreasing intervals are the x-ranges between (and beyond) the turning points.
Reading increasing/decreasing intervals
A graph has a local maximum at x = 1 and a local minimum at x = 4.
State where f is increasing and decreasing (assume f is defined for all x).
Step by step
- Increasing: graph rises left to right. This happens before x = 1 and after x = 4.
- Decreasing: graph falls left to right. This happens between the two turning points.
Final answer
Increasing for x < 1 and x > 4. Decreasing for 1 < x < 4.
Include or exclude the endpoints?: At a turning point the graph is momentarily flat — not increasing or decreasing.
IB accepts both strict (x < 1) and non-strict (x ≤ 1) inequalities here.
Use strict inequalities to be safe.
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The big idea: Two errors cost the most marks here. Always look at the graph — does the curve rise or fall as you move right?
The two most common errors
- Giving a single x-value instead of an interval
- Mixing up which side is increasing vs decreasing
Exam tips
- Increasing or decreasing is a range of x-values, not one point: f is increasing for x < 2.
- Use x-values, not y-values.
- Write the interval as an inequality: decreasing for 1 < x < 4.
IB question patterns to recognise
- 'State the interval(s) where f is increasing' — give x-range(s) using inequalities
- 'For what values of x is f decreasing?' — same format
- 'Describe the behaviour of f for x > 5' — state whether it is increasing, decreasing or approaching an asymptote
IB-style question — increasing/decreasing in context [4 marks]
A start-up's monthly profit, P thousand dollars, t months after launch is modelled by P(t) = −t² + 8t − 7, for 0 ≤ t ≤ 8.
(a) Find the month in which the profit is greatest.
(b) State the interval of months during which the profit is decreasing.
Step by step
- The profit curve is a downward parabola, so it rises to a single peak (the maximum) and then falls. Find the peak with the GDC, or from the vertex.
- (a) The greatest profit occurs at the turning point.
- (b) After the peak the profit falls. Decreasing runs from the maximum to the end of the domain.
- Give the decreasing interval as a range of t-values, and tie it back to the context — profit is falling from month 4 onward.
Final answer
(a) Month 4. (b) Profit is decreasing for 4 < t ≤ 8.
The big idea: When a function models a real situation, an increasing interval means the quantity is growing and a decreasing interval means it is shrinking.
Always interpret in the context of the problem.
Contextual interpretation
The temperature in a room T(t) is modelled over 10 hours.
The graph shows T increases for 0 < t < 3, then decreases for 3 < t < 10.
What does this tell us?
Step by step
- Increasing interval: 0 < t < 3 → temperature is rising for the first 3 hours.
- Decreasing interval: 3 < t < 10 → temperature is falling from hour 3 to hour 10.
- Turning point at t = 3 → the room was at its hottest after 3 hours.
Final answer
The room heats up for 3 hours then cools down for 7 hours. Maximum temperature occurs at t = 3.
State increasing/decreasing and interpret: IB context questions often ask you to do both: (1) identify the interval and (2) explain what it means.
Answer in two sentences: 'f is increasing for 0 < x < 3.
This means the temperature is rising during the first 3 hours.'